Question

Cost of Driving a Car The annual fixed costs of owning a small sedan are 4252 dollar, assuming the car is completely paid for. The cost to drive the car is approximately 0.14 dollar per mile. Write a linear equation that relates the cost \(C\) and the number \(x\) of miles driven annually. Source: \(A A A, 2017\)

Short answer

The linear equation is \C = 4252 + 0.14x\.

Step-by-step solution

Understand Fixed Costs

The fixed costs are the costs that do not change with the number of miles driven. Here, the annual fixed costs are given as 4252 dollars.

Understand Variable Costs

The variable cost is the cost per mile driven. In this case, it is given as 0.14 dollars per mile.

Define Variables

Let the total cost be represented by the variable \(C\) and the number of miles driven annually be represented by the variable \(x\).

Write the Linear Equation

Combine the fixed and variable costs into a linear equation. The form of the equation is: \[C = \text{fixed costs} + (\text{variable cost per mile}) \times (\text{number of miles}) \]Substitute the given values:\[C = 4252 + 0.14x\]

Key concepts

Understanding Fixed Costs

Fixed costs are expenses that do not change regardless of how much you use your car. When owning a small sedan, these costs could include insurance, taxes, and loan payments, but assuming the car is paid for, these might consist of registration fees and other essential services. In the example provided, the fixed costs amount to 4252 dollars annually. This means, even if you don't drive a single mile in a year, you will still incur this cost.

Fixed costs are crucial in understanding the overall cost because they are constants. No matter how much or how little you drive, this component of the total cost remains unchanged.

Fixed costs allow car owners to budget better as they provide a predictable expense. While they might vary year to year due to inflation or changes in insurance policy, within a single year, they stay the same.

Knowing your fixed costs can help in making decisions such as whether to keep an older car or buy a newer, more efficient model.

Understanding Variable Costs

Variable costs, on the other hand, are expenses that change depending on how much you drive your car. These costs are directly proportional to the number of miles driven. In our example, the cost per mile driven is given as 0.14 dollars.

These costs typically include fuel, maintenance, and wear-and-tear. The more you drive, the more often you need to refuel and service your car, and the faster parts like tires and brakes wear out.

Knowing the variable cost per mile helps in understanding how economical driving your car is. It allows you to calculate the marginal cost of additional miles and make more informed decisions, like whether to drive, use public transport, or consider car-sharing options.

To get a rough estimate of your variable costs, record your expenses over a period and divide that by the miles driven. This not only helps with budgeting but also with identifying ways to reduce costs.

Calculating the Cost of Driving

The cost of driving encompasses both fixed and variable costs. It's essential to understand this to get a holistic view of your travel expenses. We combine both types of costs to form a linear equation.

Let's break it down:

• The fixed costs, which are constant regardless of mileage, amount to 4252 dollars annually.
• The variable costs increase with each mile driven and are 0.14 dollars per mile.
• We represent the total cost by the variable \(C\) and the number of miles driven annually by \(x\).

The linear equation combining both fixed and variable costs is:
\[C = 4252 + 0.14x\]

Here, the term \(4252\) is the fixed cost component, and \(0.14x\) represents the variable cost per mile multiplied by the number of miles driven. This equation helps you calculate the total cost of driving for any number of miles \(x\).

For example, if you drive 10,000 miles in a year, substituting \(x = 10000\) into the equation calculates the total annual cost:\[C = 4252 + 0.14 \times 10000\]. By solving this, you can plan your budget more effectively and understand the financial impact of your driving habits.